Sleep means to rest your mind and body, usually at night. So it should be stative. But we often say something like "I am sleeping". So it looks also like a dynamic verb. Is sleep stative or dynamic or both?
Learn English – Is “sleep” stative or dynamic
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"He plays the piano" can be stative, when for example, it is being used to state a fact:
A: Does he have any hobbies?
B: He plays the piano.
Here, "He plays the piano" is stative.
"He plays the piano" can be dynamic, when describing someone actually doing it:
He rises, goes to the piano, gravely sits down, then, plays the piano.
"he plays the piano" in this context, is dynamic.
Now, is the following "have" stative or dynamic?
(3) I have to go.
(3a) I have got to go.
(3b) I'm having to go.
It seems that -- if I'm understanding CGEL right -- that your examples are all using the stative "HAVE" (not the dynamic "HAVE"). And that it is a lexical verb in your 1st and 3rd versions (#3 and #3b), while it is an auxiliary verb in your 2nd version (#3a) due to the "have got".
I intend to show some excerpts from CGEL that is directly related to this topic, but first I'll have to provide some background info (that's also in CGEL). For instance, some background info on the properties of auxiliary verbs and modal auxiliary verbs.
Auxiliary verbs have syntactic properties that distinguish them from the open class of lexical verbs; and modal auxiliaries have an additional set of properties that distinguish them from the other auxiliaries (CGEL page 92). A quick summary and examples of these properties follow, which use "WILL" for the examples. CGEL page 108:
[48] - - - - - AUXILIARY PROPERTIES
[A] Primary verb negation - - - It will not work.
[B] Subject-auxiliary inversion - - - Will it rain?
[C] Emphatic polarity - - - I WILL help you.
[D] Stranding - - - He won't attend, but I will __.
[E] Exclusion of "DO" in code - - - (*)Ed will go and I do too. ("I will go")
[F] Precede adverb/ quantifier - - - They will probably/all accept.
[G] Negative forms - - - It won't help.
[H] Reduced forms - - - She'll be here soon.
[49] - - - - - MODAL AUXILIARY PROPERTIES
[I] Only primary forms - - - (*)It's expected to will finish soon.
[J] No agreement - - - She will win. vs (*)She wills win.
[K] Only bare infinitival complement - - - It will be over. vs (*)It will to be over.
[L] Can occur in remote apodosis - - - If it weren't for her I would give up.
[M] Modally remote preterite in main clause - - - I would ask you to treat it seriously.
In general, the auxiliary verbs will have the properties [A-H], and the modal auxiliaries will have those properties and the additional properties [I-M]. Although the prototypical central modal auxiliary verbs will have the properties [A-M], there are some modal auxiliaries that don't have all of them, e.g. "must".
Let's first see some info about the dynamic "HAVE". The dynamic "HAVE" is a lexical verb (not an auxiliary). CGEL page 111:
(b) Dynamic have
This is a lexical verb in all varieties of English:
[56]
i. He had a swim. He didn't have a swim. Did he have a swim?
ii. He had it painted. He didn't' have it painted. Did he have it painted?
It has none of the auxiliary properties [A-H] -- cf. (*)He hadn't a swim, (*)Had he a swim?, etc. As the label 'dynamic' indicates, it expresses an event rather than a state. In [i] it is used as a 'light verb' (Ch. 4, &7): the main semantic content is in the following noun. Similar is its use with the meaning "experience": I didn't have any difficulty in persuading her. In [ii] it is a catenative verb taking an object and a past-participial complement; it can also take a bare infinitival: Did she have you retype it?
The constructions that involve "have to V" use the stative "HAVE"; and those usages are of a lexical verb and not an auxiliary (except for some dialects). CGEL page 111-2:
(c) Stative have and the idiom "have got"
Where have expresses a state rather than an event it is replaceable by the idiom have got (subject to conditions outlined below):
[57]
i. a. She has a swim every day. - - - b. (*)She has got a swim every day. -- [dynamic]
ii. a. She has a swimming-pool. - - - b. She has got a swimming-pool. -- [stative]
iii. a. She has to swim each day. - - - b. She has got to swim each day. -- [stative]
Stative have occurs either with an NP object, as in [ii], expressing possession and similar relations (cf. She has [many virtues / two sons]), or as a catenative verb with a to-infinitival complement, as in [iii], where the meaning is of obligation or necessity, much like that of "must" (see &9.11). Note that in spite of its semantic similarity to must this have has none of the modal properties [I-M]. For example, it has secondary forms and hence can appear in the progressive (I'm having to work late tonight), the perfect (I've had to do it all myself), and with a modal (We may have to cancel it). There is therefore no case at all for including it in the syntactic class of modal auxiliaries.
That above info about [57.iii] seems to be directly related to what you were asking about in your post (because it uses the "have to V" construction). Also, note the progressive example: "I'm having to work late tonight" (which has a form similar to your 3rd example #3b).
Within this same section of the CGEL, there is this following bit which might be helpful too. Page 112:
. . . While dynamic have is invariably a lexical verb, stative have can behave as either a lexical verb or, in some varieties, an auxiliary. This means that for the negative we have either don't have or haven't (or the analytic forms with not), and analogously with inversion. If we include have got too, we find therefore the following possibilities:
[58]
i. a. I have enough tea. - - I don't have enough tea. -- Do I have enough tea? - - (%)I haven't enough tea. - - (%)Have I enough tea?
i. b. I have got enough tea. - - I haven't got enough tea. -- Have I got enough tea?
ii. a. I have to read it all. - - I don't have to read it all. -- Do I have to read it all? - - (%)I haven't to read it all. - - (%)Have I to read it all?
ii. b. I have got to read it all. - - I haven't got to read it all. - - Have I got to read it all?
This following excerpt also has info related to the OP's topic. On page 113:
Stative have: lexical or auxiliary (doesn't have vs hasn't)
In AmE stative have always behaves as a lexical verb (and is preferred over have got). In BrE the lexical use has become common too, and the auxiliary use is tending to sound relatively formal or old-fashioned (with have got or lexical have preferred). The auxiliary use is hardly possible with a habitual interpretation: Have you to mow the lawn?, for example, shows the same restriction as have got in [60.i]. (F.E.: [60.i] I've got to mow the lawn. - single obligation)
And then there's the topic of modality, where "HAVE" is involved too, as a lexical modal (and as an auxiliary in the idiom "have got to V"); and there's also a "dynamic" term involved: deontic vs dynamic vs epistemic. CGEL page 205-6:
9.11 Lexical modals have (got), and quasi-modal be
(a) Strong modality
Have and have got
[66]
i. You have (got) to come in now. - - [deontic]
ii. Now that she has lost her job she has (got) to live extremely frugally. - - [dynamic]
iii. This has (got) to be the worst restaurant in town. - - [epistemic]
Have and have got are most commonly used for deontic necessity, as in [i]. Here they characteristically differ from must in being objective rather than subjective: with [i] I'm likely to be relaying someone else's instruction but with You must come in it's more likely that I am myself telling you to. With dynamic necessity, as in [ii], they are more likely than must. The epistemic use of [iii] is widely found in AmE, but is still fairly rare in BrE. The differences between have and have got are dealt with in &2.5.6. (fn 65)
In that above excerpt, those uses of "HAVE" are of the stative "HAVE" (especially since they have alternations with "have got"), and they all involve the construction(s) "have (got) to V".
So in conclusion, if I'm understanding all this stuff in this post properly, it seems that the OP's three examples (#3, #3a, #3b) are all using the stative "HAVE"; and that those versions that use "HAVE got", the "HAVE" is an auxiliary verb, otherwise it is a lexical verb (and also a lexical modal). And all three involve strong modality due to using either the lexical modal "HAVE" or the idiom "have got" in their "have (got) to V" constructions.
NOTE: CGEL is the 2002 reference grammar by Huddleston and Pullum et al., The Cambridge Grammar of the English Language (CGEL).
Best Answer
Don't think of the names of the categories quite so literally. The University of Victoria's Study Zone specifically lists "sleep" as a dynamic verb, and from the description it should be pretty clear why: